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Magazine Chapter 14: Problem 92
Solve for \(P: A=\frac{P t}{P+t}\)
Short Answer
Expert verified
\(P = \frac{-At}{A - t}\)
Step by step solution
01
Isolate the Fraction
In order to make it easier to solve for P, first multiply both sides by (P+t) to clear the fraction. This gives: A(P+t) = Pt.
02
Distribute A
Next, distribute A across the terms inside the parenthesis on the left side. This gives: AP + At = Pt.
03
Rearrange to Solve for P
Rearrange terms to isolate P on one side of the equation: AP - Pt = -At. This would allow us to factor out P from the left side of the equation.
04
Factor Out P
Now, take out P as a factor and rewrite the equation as: P(A - t) = -At.
05
Divide by (A-t)
Finally, to solve for P, divide both sides by (A - t) to get P on its own. This gives the final answer: \(P = \frac{-At}{A - t}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Fraction Clearing
Fraction clearing is an essential step in solving algebraic equations that include fractions. The goal is to remove the fraction by multiplying each side of the equation by the denominator. This simplifies the equation, making it easier to work with.
In the exercise, the equation was initially given as \(A = \frac{Pt}{P+t}\). To clear the fraction, we multiply both sides by \(P+t\) to remove the denominator. This results in \(A(P+t) = Pt\). The expression \(P+t\) is canceled out from the denominator, leaving a simple algebraic equation where both sides are free of fractions.
In the exercise, the equation was initially given as \(A = \frac{Pt}{P+t}\). To clear the fraction, we multiply both sides by \(P+t\) to remove the denominator. This results in \(A(P+t) = Pt\). The expression \(P+t\) is canceled out from the denominator, leaving a simple algebraic equation where both sides are free of fractions.
- Clearing fractions helps to reduce complexity in solve-algebraic equations.
- Ensures you have integer coefficients which are easier to manipulate.
Distributive Property
The distributive property is a crucial algebraic principle used to break down expressions. When you distribute a multiplication over an addition or subtraction inside parentheses, you ensure each term is multiplied by the factor outside the parentheses.
In our example, after clearing the fraction, we have \(A(P+t) = Pt\). Next, the distributive property allows us to expand this to \(AP + At = Pt\). This ensures each part of \(P+t\) is multiplied by \(A\).
In our example, after clearing the fraction, we have \(A(P+t) = Pt\). Next, the distributive property allows us to expand this to \(AP + At = Pt\). This ensures each part of \(P+t\) is multiplied by \(A\).
- Makes equations simpler by spreading out multiplication across terms.
- Assists in unlocking the structure of an equation for further simplification.
Factoring
Factoring is a method used to simplify expressions or solve equations, where you express a term as a product of its factors. This technique often helps in revealing solutions and simplifying the problem.
Once we have the equation rearranged to \(AP - Pt = -At\), factoring allows us to "factor out" the common variable, \(P\), from the equation. This is done by recognizing \(P\) is common to both terms and rewriting the equation as \(P(A - t) = -At\). Now, \(P\) is isolated with a simple expression \((A-t)\), simplifying the path towards its solution.
Once we have the equation rearranged to \(AP - Pt = -At\), factoring allows us to "factor out" the common variable, \(P\), from the equation. This is done by recognizing \(P\) is common to both terms and rewriting the equation as \(P(A - t) = -At\). Now, \(P\) is isolated with a simple expression \((A-t)\), simplifying the path towards its solution.
- Can turn complex expressions into more friendly forms.
- Essential for isolating variables in algebra.
Isolation of Variable
The final goal in solving an equation is often to isolate the variable of interest. When the variable is isolated, it appears alone on one side of the equation, signifying it has been solved for.
The last steps of our solution involve isolating \(P\). After factoring, we have \(P(A - t) = -At\). To isolate \(P\), divide both sides by \(A-t\), giving \(P = \frac{-At}{A-t}\). Now, \(P\) stands alone on one side of the equation, providing the final solution.
The last steps of our solution involve isolating \(P\). After factoring, we have \(P(A - t) = -At\). To isolate \(P\), divide both sides by \(A-t\), giving \(P = \frac{-At}{A-t}\). Now, \(P\) stands alone on one side of the equation, providing the final solution.
- Helps present the variable explicitly in terms of other constants or variables.
- Results in a clear, straightforward solution suitable for interpretation.
